Apply armature changes to mesh so that both the mesh and armature look the same in pose and edit mode
The problem lies in the fact that when you selected all bones of the rig, you didn't select all "Machinary" bones, hidden in others bone layers. Go tot the properties tab, armature, bone layers, turn on every single layer, THEN check if any other bone is hidden (ALT H), then select all and apply pose as rest pose.

In my example picutre a classic Rigify rig, and the same rig with all the machinary bones revealed.



If that doesn't solve the problem, after you have applied your pose as rest pose, select the "bn chest" bone, in the bone constraints panel search for "stretch to" constraint and click the "RESET" button. Do the same with bn neck 01 and bn neck 02.